Malmquist-Type Theorems for Cubic Hamiltonians

The aim of this paper is to classify the cubic polynomials H(z,x,y)=(j+k <= 3)Sigma -a(jk)(z)x(j)y(k) over the field of algebraic functions such that the corresponding Hamiltonian system x' = H-y, y' = - H-x has at least one transcendental algebroid solution. Ignoring trivial subcases, the investigations essentially lead to several non-trivial Hamiltonians which are closely related to Painleve's equations P-I, P-II, P-34, and P-IV. Up to normalisation of the leading coefficients, common Hamiltonians are HI : = -2y(3) + 1/2x(2) - zy H-II/34 : x(2)y - 1/2y(2) + 1/2zy +kappa x x(2)y + xy(2) +2zxy+2 kappa x+2 lambda y HIV: 1/3 (x(3)+y(3))+zxy+kappa x+lambda y, but the zoo of non-equivalent Hamiltonians turns out to be much larger.